In practice, you would use pre-made tables of standard terms like e^(a*t), t, sin(a*t) and such. You then transform your function into the sum of terms given in the table and then substitute time-domain terms into s-domain.

E.g.

f(t) = sin(2*t)*cos(3*t)

f(t) = 0.5 * sin(5*t) - 0.5*sin(t)

as Laplace(sin(a*t)) = a/(s^2+a^2)

F(s) = Laplace(f(t)) = 0.5*5/(s^2+25)-0.5/(s^2+1)

You can collect the terms to make it into a single function which you can then plot against s.

If your function can not be disassembled into terms found in tables of Laplace transform, well, then you'll have to calculate above integral transform by hand.

Hope I helped

P.S. most of calculus or engineering math handbooks have Laplace transform tables in their appendices.